Rt di COVID-19 stimato per le province italiane.
Max Pierini, Alessio Pamovio & NOTIZIÆ Telegram Channel
A simple method is presented to estimate effective reproduction number $R_t$ of COVID-19 in italian provinces with a Markov chain Monte Carlo and Poisson likelihood parametrized on daily new cases.
New cases $y_r$ for each $r$ italian province (source Dipartimento Protezione Civile), will be smoothed with rolling mean (gaussian, window 7, std 2). Smoothed new cases will be adjusted to be $>0$ to avoid negative values (due to data error corrections).
For each day $t$, smoothed new cases $y_{r,t}$ will be supposed distributed as Poisson with $\lambda_{r,t}$ parameter
$$ y_{r,t} \sim \mathcal{P}(\lambda_{r,t}) $$where $\lambda_{r,t}$ is defined by the serial interval inverse $\gamma$, previous day smoothed new cases $k_{r,t-1}$ and effective reproduction number in time $R_{r,t}$ (ref: Bettencourt & Ribeiro 2008)
$$ \lambda_{r,t} = k_{r,t-1} e^{\gamma (R_{r,t} - 1)} $$Parameters $R_{{p,t}}$, for each province $p$ and time $t$ (day) will be distributed as half normal with mean equal to previous day posteriors $R_{{p,t-1}}$ and precision $\tau$
$$ R_{{p,t}} \sim \mathcal{{N}}^+(R_{{p,t-1}}, \tau) $$where, first day $R_{{p,0}}$ (outcome) is set to zero
$$ R_{{p,0}} = 0 $$and $\tau$ is defined as $1 / \sigma^2$, where $\sigma$ is the standard deviation of $R_t$ priors, previuosly estimated for italian provinces (using the same method but with overarching uninformative prior distribution of $\tau$) as
$$ \sigma \sim \mathcal{N}(\mu_{=0.299}, \sigma_{=0.004}) $$If previous new cases are zero $k_{p,t-1}=0$, parameter $R_{p,t}$ is undefined, given the chosen function for $\lambda_{p,t}$ parameter of Poisson likelihood, even if it should be $R_{p,t}=0$ (no new cases means null effective reproduction number). Thus, in these cases, priors of $R_{p,t}$ will be forced to
$$ R_{p,t} \sim \mathcal{N}^+(0, \tau) \;,\; k_{p,t-1}=0 $$and previous new cases will be forced to $k_{p,t-1}=1$, so that $\lambda_{p,t}$ will be
$$ \lambda_{p,t} = e^{\gamma( \mathcal{N}^+(0, \tau) - 1 )} \;,\; k_{p,t-1}=0 $$Exported from Italia/Rt_Province.ipynb committed by MaxDevBlock on Sun Nov 8 15:07:08 2020 revision 1, 19c0d10